Complete Proceedings (21 Mb) more >>
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1300 | Lunch |
1400 |
Arrival and registration |
1630 | Invitation coffee break |
Chairmans, Daniela Velichová, Miroslav Lávička | |
1700 |
Daniela Velichová: 30 Years of Seminars on Geometry and Graphics in Slovakia |
1720 |
Invited lecture: prof. RND. Ján Čižmár, PhD. |
Poster section | |
Margita Vajsáblová: Geometric Aspects in Mathematical Foundations of Cartography | |
1800 |
Dinner |
1900 |
Get-together party |
800 | Breakfast |
Chairman: Pavel Pech | |
900 | Invited lecture: RNDr. Michal Bizzarri, PhD. Geometry and Tool Motion Planning for Curvature Adapted CNC Machining |
1000 | Coffee break |
Chairwoman: Daniela Richtáriková | |
1030 | Hellmuth Stachel: Billiard Motions in Ellipses -- Invariants of Projective Nature |
1050 | Pavel Pech: Chord of Conics |
1110 | Miroslav Lávička: Note on Curves of Tschirnhaus Type |
1130 | Jan Vršek: Symmetries of Discrete Curves Via Trigonometric Interpolation |
1200 | Lunch |
Chairman: Pavel Chalmovianský | |
1400 |
Invited lecture: Mgr. Michal Piovarči, PhD. |
1500 | Alexej Kolcun: Fergusonova krivka v príkladoch |
1520 | Marta Hlavová: Interpolation of Transcendental Curve Points |
1540 | Marcel Makovník: Mesh Refinement Near Singularities |
1600 | Coffee break |
Chairman: Jan Safařík | |
1630 | Dana Kolářová: Plocha šikmého průchodu v nosné konstrukci historických mostů |
1650 | Tatiana Rückschlossová, Margita Vajsáblová: Geometria v profile absolventa stavebného odboru do roku 2030 |
1710 | Michaela Holešová: Constructions of Quatrefoil |
1800 |
Dinner |
1900 |
Meetings of societies SSGG, ČSGG |
800 | Breakfast |
Chairman: Jan Vršek | |
900 | Pavel Chalmovianský: What is a Curve? |
920 | Alžbeta Mackovová: Voronoi Diagram in Three-Dimensional Hyperbolic Space |
940 | Adriana Bosáková: A Note on the Structure of the Intersection Multiplicity of Two Plane curves |
1000 |
Coffee break |
Chairwoman: Margita Vajsáblová | |
1030 |
Invited lecture: prof. Gunter Weiss |
1130 | Michal Zamboj: Twisted Filaments with Polyhedral Symmetries |
1200 | Lunch |
1300 | Excursion |
1900 |
Conference dinner |
800 | Breakfast |
GeoGebra workshop | |
Chairmans: Michaela Holešová, Roman Hašek | |
900 | Věra Ferdiánová: GeoGebra nástroje ve výuce planimetrie |
920 | Marie Chodorová: Kolineace kuželoseček v GeoGebře |
940 | Jakub Řada: GeoGebra Tools for Drawing in Double Orthogonal Projection and 4D Perspective |
1000 | Roman Hašek: Dynamic Geometry Online |
1020 | Jan Šafařík, Jana Bulantová, Lucie Zrůstová: Sbírka řešených příkladů z konstruktivní geometrie |
1130 | Lunch |
1300 | Departure |
Geometry and Tool Motion Planning for Curvature Adapted CNC MachiningBIZZARRI Michal Katedra matematiky Fakulta aplikovaných věd, Západočeská univerzita v Plzni CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between “dual” classical results in surface theory concerning osculating circles of surface curves and osculating cones of tangentially circumscribed developable surfaces. Practically, it serves as an effective basis for tool motion planning. Unlike previous approaches to curvature-adapted machining, we solve locally optimal tool positioning and motion planning within a single optimization framework and achieve curvature adaptation even for convex surfaces. This is possible with a toroidal cutter that contains a negatively curved cutting area. The effectiveness of our approach is verified at hand of digital models, simulations and machined parts, including a comparison to results generated with commercial software. |
A Note on the Structure of the Intersection Multiplicity of Two Plane curvesBOSÁKOVÁ Adriana Fakulta matematiky, fyziky a informatiky, Univerzita Komenského v Bratislave Intersection multiplicity is an invariant in algebraic geometry, which describes the complexity of an intersection of algebraic varieties. Studying its internal structure can help us describe the algebraic and geometric properties of such intersection. We focus on the case of the intersection of two plane affine algebraic curves. By existing results, we know that the intersection multiplicity is greater or equal to (mn + t), where m and n are the multiplicities of the intersection point on the individual curves and t is the number of their common tangents at this point. In this paper we show an improved version of this result. |
Sbírka řešených příkladů z konstruktivní geometrieBULANTOVÁ Jana, ŠAFAŘÍK Jan, ZRŮSTOVÁ Lucie FAST VUT Brno Cílem příspěvku je seznámit s novou online sbírkou řešených příkladů z konstruktivní geometrie, která vznikla během distanční výuky v AR 2020/2021 na Stavební fakultě VUT v Brně především za pomocí GeoGebry, počítačového programu pro interaktivní geometrii. |
What is a Curve?CHALMOVIANSKÝ Pavel Department of Algebra and Geometry, Faculty of Mathematics and Physics, Comenius University We look at the notion of curve from different points of view. |
Kolineace kuželoseček v GeoGebře.CHODOROVÁ Marie Katedra algebry a geometrie, Přírodovědecká fakulta Univerzity Palackého v Olomouci Cílem příspěvku je ukázat na příkladech konstrukce kuželoseček pomocí kolineace, kdy kolineákním obrazem kuželosečky je kružnice (příp. jiná kuželosečka). Tyto konstrukce mají praktické využití při sestrojování řezů rotačních a zejména nerotačních kuželů. Konstrukce jsou zpracovány v GeoGebře formou pracovních listů. |
Aktuálny odkaz Euklidových ZákladovČIŽMÁR Ján Príspevok uvádza krátky prehľad obsahu trinástich kníh kardinálneho Euklidovho diela Základy (Stoicheia) z hľadiska dnešnej klasifikácie matematických odborov. Dielo je po prvý raz v úplnosti pripravené do tlače v slovenskom preklade autora príspevku a je opatrené stručným komentárom približujúcim obsah diela v jazyku dnešnej matematiky. |
GeoGebra Nástroje ve výuce planimetrieFERDIÁNOVÁ Věra KMA, PřF, Ostravská univerzitaOstrava, Czech republic e-mail: vera.ferdianova@gmail.com Cílem příspěvku bude představení tvorby GeoGebra nástrojů, jenž napomáhají při běžných planimetrických konstrukcí. |
Dynamická geometrie online / Dynamic Geometry OnlineHAŠEK Roman Jihočeská univerzita v Českých Budějovicích, Pedagogická fakulta Příspěvek je zaměřen na online prezentaci matematického a geometrického obsahu. Věnuje se konkrétnímu způsobu prezentace tohoto obsahu prostřednictvím materiálů ve formátu HTML, které mohou mít jak podobu samostatné webové stránky, tak i dílčí součásti nějakého online vzdělávacího systému, například Moodle. Na vybraných příkladech bude ukázáno, jak lze použitím knihoven JavaScriptu JSXGraph a MathJax vytvořit kvalitní prezentaci geometrického obsahu, v níž se volně kombinuje matematický text ve formátu LaTeX s interaktivními dynamickými obrázky. |
Interpolation of Transcendental Curve Points/ Interpolace bodů transcendentní křivkyHLAVOVÁ Marta ČVUT v Praze, Fakulta strojní This contribution deals with the dependence of geometrical accuracy of a natural cubic spline on the Euclidean distance between the definition points. As an example, a transcendental curve - a graph of function f: y = sin(x), x from the interval [0,2Pi] - is used. |
Constructions of QuatrefoilHOLEŠOVÁ Michaela KSMAM ŽU v Žiline In architecture, we meet with different and also more complex shapes. Due to the technical construction, they are usually made up of basic geometric elements, such as a line and a circle. They are used to create interesting floor plans, openings and decorative elements. In addition to the relatively frequently used ovals, we also encounter the so-called quatrefoil based on four circles or ovals in Slovak architecture. These are arranged symmetrically and are connected to each other by lines which are predominantly tangential to these circles. We will show some interesting constructions of quatrefoils. |
Plocha šikmého průchodu v nosné konstrukci historických mostůKOLÁŘOVÁ Dana Fakulta architektury ČVUT v Praze Příspěvek představuje vývoj použití plochy šikmého průchodu v nosné konstrukci historických mostů. |
Fergusonova krivka v príkladoch / Ferguson Curve in the ExamplesKOLCUN Alexej Katedra informatiky a počítačů Přírodovědecká fakulta Ostravská univerzita Pri demonštrácií vzájomného vzťahu riadiacich prvkov a výsledného tvaru parametrických kriviek sa učebnice spravidla obmedzujú na kvalitatívnu charakterizáciu tohto vzťahu. V príspevku formulujeme niekoľko jednoduchých úloh, kde pre Fergusonovu krivku nachádzame detailnejšie – kvantitatívne vzťahy medzi riadiacimi elementami a výsledným tvarom krivky. |
Note on Curves of Tschirnhaus TypeLÁVIČKA Miroslav Katedra matematiky Fakulta aplikovaných věd Západočeská univerzita v Plzni Special Pythagorean hodograph curves which can be considered, from construction point of view, as degree n generalizations of the famous Tschirnhaus cubic will be presented. Their properties will be discussed. In addition, it will be proved that for each n there exists only one curve of Tschirnhaus type up to similarities. |
Voronoi Diagram in Three-Dimensional Hyperbolic SpaceMACKOVOVÁ Alžbeta Katedra algebry a geometrie, Fakulta matematiky, fyziky a informatiky, Univerzita Komenského v Bratislave Voronoi diagrams belong to favorite structures in computational geometry. In our work, we focus on construction and properties of Voronoi diagram in three-dimensional hyperbolic space represented by Poincaré ball model. We first define and illustrate the basic geometric elements of Poicaré ball model that we will work with. Finally, we construct a Voronoi diagram and describe some of its properties, which depend on the position of the generators. |
Mesh Refinement Near SingularitiesMAKOVNÍK Marcel Katedra algebry a geometrie Fakulta matematiky, fyziky a informatiky Univerzita Komenského v Bratislave A singular point of an implicit surface is a point, whose gradient is a zero vector. Mesh refinement via quadric fitting takes point-normal pairs as an input. Hence, singularities in the mesh may be obtained by assigning zero normal vector to the input point. In this paper we inspect on the behaviour of the refinement near the singularity with respect to connectivity of the mesh. We provide examples of refinement on synthetic and real data. |
Chord of ConicsPECH Pavel Jihočeská univerzita v Č. Budějovicích, Pedagogická fakulta The paper deals with locus of points related to chords of conic sections. Firstly the locus is explored using dynamic geometry software GeoGebra, secondly using |
Perception-Aware Computational FabricationPIOVARČI Michal Institute of Science and Technology Austria Haptic and visual feedback are important for assessing objects´ quality and affordance. One of the benefits of additive manufacturing is that it enables the creation of objects with personalized tactile and visual properties. This personalization is realized by the ability to deposit functionally graded materials at microscopic resolution. However, faithfully reproducing real-world objects on a 3D printer is a challenging endeavor. A large number of available materials and freedom in material deposition make exploring the space of printable objects difficult. Furthermore, current 3D printers can perfectly capture only a small amount of objects from the real world which makes high-quality reproductions challenging. Interestingly, similar to the manufacturing hardware, our senses of touch and sight have inborn limitations given by biological constraints. In this talk, I will demonstrate how we can leverage the limitations of the Human Sensorial System to increase the apparent gamut of a 3D printer by combining numerical optimization with perceptual insights. Instead of optimizing for exact replicas, we search for perceptually equivalent solutions. I will show applications of the proposed methodology to designing objects with prescribed compliance, mimicking the haptics of drawing tools, and manufacturing objects with spatially varying gloss. |
GeoGebra Tools for Drawing in Double Orthogonal Projection and 4D PerspectiveŘADA Jakub Matematický ústav UK, Matematicko-fyzikální fakulta, Univerzita Karlova, Praha With the advent of modern technology, it is often much more accessible to draw on a computer than on paper. It is necessary to choose the right program. There are efforts to make special software for specific projection, but usually, these programs do not last long. For example, Petr Plavjanik´s program - Deskriptivní geometrie. In 1999, the author celebrated success with it in SPA and INVEX ´99. After a few years, its development was stopped. For this reason, it is better and more convenient for the users to use some general graphic software. GeoGebra is one of them, and it also has an outstanding possibility to create your own custom tools. Each projection has its principles for which custom tools can be created. This article aims to show using GeoGebra with own custom tools as additional software to display objects in double orthogonal projection and 4D perspective. |
Geometria v profile absolventa stavebného odboru do roku 2030RÜCKSCHLOSSOVÁ Tatiana Katedra matematiky a deskriptívnej geometrie, Stavebná fakulta STU v Bratislave Študijné programy na Stavebnej fakulte, Fakulte architektúry a dizajnu a na Ústave manažmentu STU v Bratislave sú po akreditácii 2021 naďalej otvorené geometrickým predmetom. Stavebný inžinier, architekt, či geodet budúcnosti má veľké softvérové a technologické možnosti v tvorbe a realizácii stavebného objektu, pričom geometrická a priestorová predstavivosť umocňuje ich efektívne využitie. Obsah, metódy a formy výučby geometrie si vyžadujú aktualizáciu súvisiacu s následnou aplikáciou vedomostí študenta v odborných predmetoch a v stavebnej praxi. V príspevku ukážeme tieto aspekty na aplikáciách geometrie vo výučbe geometrických predmetov pre konkrétne študijné odbory. Zároveň uvedieme analýzu, skúsenosti a ukážky metód dištančnej formy výučby geometrických predmetov. |
Billiard Motions in Ellipses -- Invariants of Projective NatureSTACHEL Hellmuth Vienna University of Technology Institute of Discrete Mathematics and Geometry A billiard is the trajectory of a mass point in a domain with ideal physical reflections in the boundary. Already for two centuries, billiards in ellipses have attracted the attention of mathematicians, beginning with J.-V. Poncelet and C.G.J. Jacobi. Recent computer animations, which were carried out by D. Reznik (Brazil), stimulated a new vivid interest on this well studied objects. |
Geometric Aspects in Mathematical Foundations of CartographyVAJSÁBLOVÁ Margita Katedra matematiky a deskriptívnej geometrie, Stavebná fakulta STU v Bratislave Cartographic projection represents the relationship between the reference surface of the Earth and its map image. In essence, it has a geometric and mathematical character. In this paper, the emphasis is on the presentation of geometric aspects in the selection and creation of cartographic projection, namely the shape of the reference surface of the Earth, geometric characteristics of the projected area, curves on reference surfaces and image shape requirements of projected elements. These aspects are included, although often hidden in the university textbook Mathematical Foundations of Cartography, in the creation of which I applied my mathematical logic, geometric eyes and cartographic heart. |
30 Years of Seminars on Geometry and Graphics in SlovakiaVELICHOVÁ Daniela ÚMF SjF STU A brief look back and an overview of the 30-years history of professional seminars on geometry and graphics held in Slovakia from the year 1991 is presented in this article. Seminars have developed into the international Symposiums on Computer Geometry and they were organized by the Slovak Society for Geometry and Graphics until 2014. These international conferences were joined from the year 2015 with the Conference on Geometry and Graphics held annually in the Czech Republic by the Czech Society for Geometry and Graphics. Successful 6 meetings of Slovak and Czech scientific communities active in geometry and computer graphics organised alternately in the Czech Republic and Slovakia proved that this idea was useful and fruitful. The 7th common Slovak-Czech Conference on Geometry and Graphics took place in Kočovce, Slovakia, in September 2021. |
Symmetries of Discrete Curves Via Trigonometric InterpolationVRŠEK Jan Západočeská univerzita v PlzniPlzeň, Česká republika e-mail: vrsekjan@kma.zcu.cz The talk will be devoted to the computation of symmetries of a~closed discrete curve in Euclidean plane. We will show how to replace discrete curve by the so called trigonometric curve, i.e., curve possessing a parametrization in terms of sines and cosines. The curve is chosen so that its symmetry group contains the symmetry group of the original discrete curve. We demonstrate that the symmetries of a trigonometric curve can be found easily, which we use to solve the original problem. We conclude the talk by observing how convex hulls, together with the previous approach, allow us to find symmetries of clouds of points. |
Geometric Problems that Take Possession of MeWEISS Gunter retired The lecture deals with my way of doing research: It often starts with getting stimulated by an article or a lecture at a conference. Putting one of its details in another context, mostly a projective geometric one, and then, by looking for meaningful generalisations, it is finally possible to extract a general principle which brings together divergent and often well-known facts. The resulting aha moments are almost addictive. Examples of such topics will be “polyhedrons made by setsquares”, “C^r-biarcs of circles and conics”, “Stellae octangulae”, “Frégier’s theorem”, “Minkowski Geometry resp. metric planes”. There remain still many open questions, which could be perhaps answered by researchers having another scientific background than mine. Unfortunately the mentioned topics are not within the actual scientific mainstream, but their inner beauty justifies to be curious and to struggle for getting answers to open questions. |
Twisted Filaments with Polyhedral SymmetriesZAMBOJ Michal Department of Mathematics and Mathematical Education, Faculty of Education, Charles University Twisted filaments are common structures in nature. We describe a geometric method of their creation, such that they possess a predefined polyhedral symmetry. We start from a well-chosen polyhedron mapped to a circumscribed 2-sphere. The images of the vertices on the 2-sphere create circular fibers in a 3-sphere in the Hopf fibration. Moreover, circles around the vertices form torus filaments around the fibers. After all, we visualize the filaments inside of the 3-sphere in a double-orthogonal and stereographic projection. |